A two-dimensional generating system modeling growth by binary cell division

We first introduce informally our grammatical scheme for modeling the generation or growth of data structures which can be viewed as the neighbor relations (topologically invariant) for planar maps, permitting binary division of "countries" or "cells", motivated by biological considerations and by the desire to generate successive patterns in the Lindenmayer fashion (i.e., by simultaneous application of rules to every cell). We give simple examples illustrating application of the rules of such grammars, and make further remarks in the biological context. Next we present formal definitions for our 2-dimensional grammars. Finally, we give some formal results for our systems; for convenience, the terminology is often that of maps and countries (rather than patterns or organisms and cells).